Leonard pairs having specified end-entries

Abstract: Fix an algebraically closed field $\mathbb{F}$ and an integer $d \geq 3$. Let $V$ be a vector space over $\mathbb{F}$ with dimension $d+1$. A Leonard pair on $V$ is an ordered pair of diagonalizable linear transformations $A: V \to V$ and $A^* : V \to V$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Let ${v_i}{i=0}^d$ (resp.\ ${v^*_i}{i=0}^d$) be such an eigenbasis for $A$ (resp.\ $A^*$). For $0 \leq i \leq d$ define a linear transformation $E_i : V \to V$ such that $E_i v_i=v_i$ and $E_i v_j =0$ if $j \neq i$ $(0 \leq j \leq d)$. Define $E^*_i : V \to V$ in a similar way. The sequence $\Phi =(A, {E_i}{i=0}^d, A^*, {E^*_i}{i=0}^d)$ is called a Leonard system on $V$ with diameter $d$. With respect to the basis ${v_i}{i=0}^d$, let ${\th_i}{i=0}^d$ (resp.\ ${a^*i}{i=0}^d$) be the diagonal entries of the matrix representing $A$ (resp.\ $A^*$). With respect to the basis ${v^*i}{i=0}^d$, let ${\theta^*i}{i=0}^d$ (resp.\ ${a_i}{i=0}^d$) be the diagonal entries of the matrix representing $A^*$ (resp.\ $A$). It is known that ${\theta_i}{i=0}^d$ (resp. ${\th^*i}{i=0}^d$) are mutually distinct, and the expressions $(\theta_{i-1}-\theta_{i+2})/(\theta_i-\theta_{i+1})$, $(\theta^_{i-1}-\theta^_{i+2})/(\theta^*_i - \theta^*_{i+1})$ are equal and independent of $i$ for $1 \leq i \leq d-2$. Write this common value as $\beta + 1$. In the present paper we consider the "end-entries" $\theta_0$, $\theta_d$, $\theta^*_0$, $\theta^*_d$, $a_0$, $a_d$, $a^*_0$, $a^*_d$. We prove that a Leonard system with diameter $d$ is determined up to isomorphism by its end-entries and $\beta$ if and only if either (i) $\beta \neq \pm 2$ and $q^{d-1} \neq -1$, where $\beta=q+q^{-1}$, or (ii) $\beta = \pm 2$ and $\text{Char}(\mathbb{F}) \neq 2$.